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Probability Density Function

Also PDF · Probability Density · Density Function · Joint PDF · Marginal Distribution · Wahrscheinlichkeitsdichte · Dichtefunktion

The probability density function p_X of a real-valued random variable X is the function whose integral over an interval [a, b] gives the probability that X takes a value in [a, b].

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Properties#

Normalization: $\int_{-\infty}^{\infty} p_X(x)\,\mathrm{d}x = 1$ (the event “any value” is certain), and $p_X(x) \ge 0$ .
A density is not a probability: $p_X(x)$ can exceed $1$ . Only its integral over an interval is a probability, and $P(X = x) = 0$ for every single value $x$ .
Joint PDF: two random variables $X, Y$ have a joint PDF $p_{X,Y}(x, y)$ . For a random vector $X \in \mathbb{R}^n$ , $P(X \in A) = \int_A p_X(x)\,\mathrm{d}^n x$ .
Marginal PDF: $p_Y(y) = \int_{-\infty}^{\infty} p_{Y,Z}(y, z)\,\mathrm{d}z$ , and likewise for $Z$ .
Conditional PDF: $p_{X|Y}(x|y) = \dfrac{p_{X,Y}(x, y)}{p_Y(y)}$ , short $p(x|y) = \dfrac{p(x, y)}{p(y)}$ (Conditional Probability).
Independence: $X, Y$ are independent iff $p_{X,Y}(x, y) = p_X(x)\,p_Y(y)$ .
Expectations are integrals against the PDF: $\mathbb{E}\{f(X)\} = \int f(x)\,p_X(x)\,\mathrm{d}x$ (Expected Value).

Example#

Sources#

MSI script p. 14 (§2.2), MSI Slides 02 > 2.2 Scalar Random Variables and Probability Density Functions (not published)
MSI script pp. 15–16 (marginals, §2.2.4; joint PDF of a random vector, §2.3)

Connections

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Random Variable

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VL02 – Probability and Statistics in a NutshellMSI
A real random variable is described by its Probability Density Function $p_X$ . Probabilities are integrals of it, $P(X \in [a,b]) = \int_a^b p_X(x)\,\mathrm{d}x$ . Independence and conditioning carry over from events to joint and conditiona… §
Probability Density Function: describes a scalar random variable, integrates to probabilities §
Conditional Probability
For random variables $X$ , $Y$ with joint PDF $p_{X,Y}$ , the conditional PDF of $X$ given $Y$ is §
Covariance
For two scalar random variables $Y$ , $Z$ with joint PDF $p_{Y,Z}$ and means $\mu_Y$ , $\mu_Z$ , the covariance is §
Expected Value
For a random variable $X$ with PDF $p_X$ and a function $f$ , the expectation operator is §
Normal Distribution
A scalar random variable $X$ is normally distributed (Gaussian) with parameters $\mu \in \mathbb{R}$ and $\sigma > 0$ , written $X \sim \mathcal{N}(\mu, \sigma^2)$ , if its PDF is §
Random Variable
Description by a density: a real-valued random variable is described by its Probability Density Function $p_X$ ; a random vector $X \in \mathbb{R}^n$ by a joint density on $\mathbb{R}^n$ . Two scalar random variables $Y, Z$ can be stacked in… §
Statistical Independence
Two random variables $X$ , $Y$ are independent if their joint PDF factorizes, $p_{X,Y}(x, y) = p_X(x)\,p_Y(y)$ for all $x, y$ . Otherwise they are dependent. §
Uniform Distribution
A random variable $X$ is uniformly distributed on $[a, b] \subset \mathbb{R}$ , $a < b$ , if its PDF is §
Random Variables and Distributions
A random variable models a quantity that is not known before the experiment, such as a noisy measurement. It is described by probabilities of events, or more conveniently by a PDF. Joint, marginal and conditional densities describe several… §
Probability Density Function — the description of a continuous random variable; joint, marginal, conditional §

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